If op had named it something different, and didn't use such a closely inspired logo, he probably could have kept it up. Or at least had a much stronger case of fair use. But this is always the problem with "fan appreciation" and our trademark system. In order for Waffle House to keep their trademark, they must enforce it. They cannot let a fan use/abuse it, or else it can get to the point where a judge can say it is been made generic (ex: escalator).
When I was young, I learned this lesson via a very scary letter from Warner Brothers. The lawyers aren't always trying to be dicks (though some clearly are), but they are paid to protect the brand. And someone registering a site with your brand name, putting up a logo very similar to your real logo.... is always 100% of the time going to get that C&D letter. And eventually a lawsuit if you fail to comply.
For this to be trademark infringement, wouldn't he need to be selling products or services that compete with Waffle House? I don't think this is an obvious easy win for the WH lawyers at all, apart from the fact that they would have much greater legal resources.
Basically trademarks are like a signature, they "prove" who you are doing business with.
More than risk; some of the tweets could be actual evidence of that confusion.
I think if the general branding vibe of the site was more about hurricanes and only made a reference to the "Waffle House index" via NOAA, it might have had better chances of surviving. And also a big disclaimer that there is no affiliation with WH would have also helped.
However, you should always speak to a lawyer if you get any kind of legal notice, but especially of this type. Never assume, thats how things can go sideways, even if you may initially have had standing to dispute the notice.
Always talk to a lawyer before taking any action in situations like this.
(Though its less likely, complying with such a letter can also have hidden costs, so a consultation may be a good idea even if that is your inclination, but you are usually far more likely to experience adverse consequences beyond those you voluntarily and knowingly undertake from disregarding than from complying wiht such a letter.)